Classes of unimodular abelian group matrices
Dennis A. Garbanati, Robert C. Thompson · Pacific Journal of Mathematics · 1972
Let G be a finite abelian group, let G o be the set of unimodular group matrices for G with rational integer entries, let Gi be the symmetric members of Go, and G 2 the positive definite symmetric members of Go.Let K be either Gi or G2.On K impose the equivalence relation of group matrix congruence by asserting A~B (for A,BeK) if and only if Ce Go exists such that A = CBC^~, where ^ denotes transposition.M. Newman has estimated the number of classes under this equivalence relation, when G is cyclic.In this paper his study is continued for abelian groups.As part of the results it is shown that the class number of K is always a power of two, and when K is Gi the exact value of this class number is obtained.When K is G2 an upper bound for class number is found and shown to be sharp by exhibiting an infinite class of groups for which it is achieved.